Find the value of [tex][tex]$k$[/tex][/tex] such that the polynomial [tex][tex]$k x^3+3 x^2-3$[/tex][/tex] and [tex][tex]$2 x^3-5 x+k$[/tex][/tex] both leave the same remainder when divided by [tex][tex]$(x-4)$[/tex][/tex].



Answer :

To solve this problem, we need to find the value of [tex]\( k \)[/tex] such that the remainders of the polynomials [tex]\( kx^3 + 3x^2 - 3 \)[/tex] and [tex]\( 2x^3 - 5x + k \)[/tex] when divided by [tex]\( x-4 \)[/tex] are the same.

### Step-by-Step Solution:

1. Substitute [tex]\( x = 4 \)[/tex] into both polynomials and set their remainders equal:
- When a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], the remainder is given by [tex]\( f(a) \)[/tex].

2. Calculate the remainder of [tex]\( kx^3 + 3x^2 - 3 \)[/tex] when [tex]\( x = 4 \)[/tex]:

[tex]\[ f(4) = k(4)^3 + 3(4)^2 - 3 = k \cdot 64 + 3 \cdot 16 - 3 \][/tex]

Simplify each term:

[tex]\[ f(4) = 64k + 48 - 3 \][/tex]

Combine the constants:

[tex]\[ f(4) = 64k + 45 \][/tex]

3. Calculate the remainder of [tex]\( 2x^3 - 5x + k \)[/tex] when [tex]\( x = 4 \)[/tex]:

[tex]\[ g(4) = 2(4)^3 - 5(4) + k = 2 \cdot 64 - 5 \cdot 4 + k \][/tex]

Simplify each term:

[tex]\[ g(4) = 128 - 20 + k \][/tex]

Combine the constants:

[tex]\[ g(4) = 108 + k \][/tex]

4. Set the remainders equal to each other and solve for [tex]\( k \)[/tex]:

We have:

[tex]\[ 64k + 45 = 108 + k \][/tex]

Solve for [tex]\( k \)[/tex]:

[tex]\[ 64k - k = 108 - 45 \][/tex]

Combine like terms:

[tex]\[ 63k = 63 \][/tex]

Divide both sides by 63:

[tex]\[ k = 1 \][/tex]

### Conclusion

Thus, the value of [tex]\( k \)[/tex] that ensures the remainders are the same for both polynomials when divided by [tex]\( x-4 \)[/tex] is:

[tex]\[ \boxed{1} \][/tex]